Let $B\left(X\right)$ be the algebra of all bounded linear operators in a complex Banach space $X$. We consider operators $T_1,T_2\in B\left(X\right)$ satisfying the relation ${\sigma }_{T_1}\left(x\right)={\sigma }_{T_2}\left(x\right)$ for any vector $x\in X$, where ${\sigma }_T\left(x\right)$ denotes the local spectrum of $T\in B\left(X\right)$ at the point $x\in X$. We say then that $T_1$ and $T_2$ have the same local spectra. We prove that then, under some conditions, $T_1-T_2$ is a quasinilpotent operator, that is $\parallel {(T_1-T_2)}^n{\parallel }^{1/n}\to 0$ as $n\to \infty$. Without these conditions, we describe the operators with the same local spectra only in some particular cases.

We study similarity to partial isometries in C*-algebras as well as their relationship with generalized inverses. Most of the results extend some recent results regarding partial isometries on Hilbert spaces. Moreover, we describe partial isometries by means of interpolation polynomials.

the existence of an $\omega$-periodic solution of the equation $\frac{{\partial }^{2}u}{\partial t^2}+\alpha \frac{{\partial }^{4}u}{\partial x^4}+\gamma \frac{{\partial }^{5}u}{\partial x^4\partial t}-\tilde{\gamma }\frac{{\partial }^{3}u}{\partial x^2\partial t}+\delta \frac{\partial u}{\partial t}-\left[\beta +\aleph {\int }_0^n{\left(\frac{\partial u}{\partial x}\right)}^2(·,\xi )d\xi +\sigma {\int }_0^n\frac{{\partial }^{2}u}{\partial x\partial t}(·,\xi )\frac{\partial u}{\partial x}(·,\xi )d\xi \right]\frac{{\partial }^{2}u}{\partial x^2}=f$ sarisfying the boundary conditions $u(t,0)=u(t,\pi )=\frac{{\partial }^{2}u}{\partial x^2}\left(t,0\right)=\frac{{\partial }^{2}u}{\partial x^2}\left(t,\pi \right)=0$ is proved for every $\omega$-periodic function $f\in C\left(\left[0,\omega \right],L_2\right)$.

Let $T,T^{\text{'}}$ be weak contractions (in the sense of Sz.-Nagy and Foiaş), $m,m^{\text{'}}$ the minimal functions of their $C_0$ parts and let $d$ be the greatest common inner divisor of $m,m^{\text{'}}$. It is proved that the space $I(T,T^{\text{'}})$ of all operators intertwining $T,T^{\text{'}}$ is reflexive if and only if the model operator $S\left(d\right)$ is reflexive. Here $S\left(d\right)$ means the compression of the unilateral shift onto the space $H^2\ominus d{H}^2$. In particular, in finite-dimensional spaces the space $I(T,T^{\text{'}})$ is reflexive if and only if all roots of the greatest common divisor of minimal polynomials...

We extend and generalize some results in local spectral theory for upper triangular operator matrices to upper triangular operator matrices with unbounded entries. Furthermore, we investigate the boundedness of the local resolvent function for operator matrices.

Let x be a positive element of an ordered Banach algebra. We prove a relationship between the spectra of x and of certain positive elements y for which either xy ≤ yx or yx ≤ xy. Furthermore, we show that the spectral radius is continuous at x, considered as an element of the set of all positive elements y ≥ x such that either xy ≤ yx or yx ≤ xy. We also show that the property ϱ(x + y) ≤ ϱ(x) + ϱ(y) of the spectral radius ϱ can be obtained for positive elements y which satisfy at least one of the...